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- Question : 2.1 - An aerospace vehicle is flying as illustrated in Fig. 2.1. If we treat the vehicle as a moving mass, energy considerations lead to the conclusion that rate of mechanical energy input storage rate of potential energy storage rate of kinetic energy where ze = Energy height = H + Show that this same result can be obtained from Eq. (2-1), which was based on force considerations. Reconcile the two approaches for the case of a constant speed climb, during which altitude changes but velocity does not.
- Question : 2.2 - An aerospace vehicle is flying as illustrated in Fig. 2.1. Suppose it is climbing much more slowly than it is moving forward, so that rc is approximately the radius of the Earth, rs. Equation (2-2) reveals that the centrifugal force associated with the flight path curvature reduces the need for lift, which, in turn, usually has the beneficial effect of reducing the drag of the lifting surfaces. Use Eq. (2-2) to show that For the representative atmosphere of Sec. 2.4.9, calculate and plot L/W as a function of flight Mach number for 0 < M0 < 25.
- Question : 2.3 - According to Ref. 2.7, the perfect gas constants used in establishing the properties of the standard atmosphere were not changed over the first 80 km (262 kft) of geometric altitude, where mixing dominates. Therefore, if the information presented in App. B is internally consistent, the functions and should be identical to 1 at any geometric altitude. Is the information of App. B internally consistent?
- Question : 2.4 - Starting from Eq. (2-8) for the behavior of atmospheric static pressure and employing the reference temperature of 400
- Question : 2.5 - One of the most commonly used hypersonic boundary layer transition prediction methods (even though it does not include many important influences and is therefore only a rule of thumb) is that 2s for freestream Mach numbers in excess of about 6. Develop an expression for this boundary similar to Eq. (2-22) and overlay the calculated result on Fig. 2.4. If this transition model is correct, are the conclusions reached in Sec. 2.4.7 still valid?
- Question : 2.6 - One of the most popular means for calculating the local heat transfer of compressible flows is the reference temperature method, which extends incompressible formulas by correcting transport properties for compressibility effects (an approximate engineering approach based on intuitive insight). 2'1,2'2 For fully turbulent hypersonic flow over a fiat plate with the wall temperature cooled to nearly the freestream static temperature, the reference temperature method would predict that the local wall heat flux where the starred transport properties are evaluated at T* = T0(1 + 0.032M20) Develop an expression for constant qw similar to Eq. (2-22) and overlay some contours of constant qw on Fig. 2.4. HAP(Trajectory) will prove useful to this exercise. If this heat transfer model is correct, what is the likelihood that the local aerodynamic heating will become a dominant factor at very high Mach numbers?
- Question : 2.7 - Employ the computer program HAP(Air) to generate the equilibrium constituents of representative air as a function of static temperature for static pressures of I and 100 arm and plot them as shown in Fig. 2.8. (a) Does this information confirm the assertion that the effect of higher static pressures is to delay the onset of dissociation and chemical reactions to higher temperatures? (b) Are these results in agreement with the information found in Figs. 2.7, 2.9, and 2.10?
- Question : 2.8 - Suppose that the mole fraction composition of air is: N2 = 78% 02 = 20% A = 1% H20 = 1% and the static pressure is 0.01 atm. Employ the computer program Hap(Equilibrium) to generate information equivalent to Figs. 2.7 through 2.10 for this mixture. Do the new constituents have a significant impact upon the equilibrium constituents or thermodynamic properties of air?
- Question : 2.9 - Employ the computer program HAP(Air) to determine the equilibrium speed of sound for representative air at a pressure of 1 atm and over the temperature range of Fig. 2.7. Compare this with the calorically perfect, nonreacting gas value calculated from Eq. (2-42) and the corresponding values of 7 and R taken from HAP(Air). What can you say generally or specifically about these results?
- Question : 2.1 - Nail down the concept of converting kinetic energy to enthalpy by calculating the difference between static and total enthalpy for a vehicle flying at 0 < M0 < 25. Use Eq. (2-47) and the representative atmosphere of Sec. 2.4.9, and find the answer in both BE and SI units.
- Question : 2.11 - Nail down the concept of isentropic compression by using Eq. (2-54) to calculate the total pressure for a vehicle flying along any constant q0 trajectory at 0 < M0 < 25. Assume 7o = 1.40.
- Question : 2.12 - For the isentropic, one-dimensional flow of a calorically perfect gas in an axial duct, use the concept of mass flow parameter (MFP) to show that the "static pressure MFP" is given by the expression Use this result to construct a chart of "static pressure MFP" as a function of Mach number for 7 = 1.40 and R = 53.3 ft .lbf/(lbm.
- Question : 2.13 - Compare the results obtained from the finite control volume and differential control volume analyses with those obtained from HAP(Gas Tables) for the following Example Cases: (a) Isentropic flow (b) Normal shock waves (c) Constant area heating (Rayleigh flow) (d) Idea/exit velocity (e) Mass flow parameter (f) Constant pressure heating
- Question : 2.14 - Consider the frictionless, one-dimensional flow of a calorically perfect gas for which there is only a heat interaction with the surroundings and the throughflow area is varied in such a way that static temperature remains constant, i.e., (a) Find an expression for the area ratio Ae/Ai in terms of 7, (Tte - Tti)/Ti, and Mi. (b) Find an expression for the total pressure ratio Pte/Pti in terms of the same variables.
- Question : 2.15 - Consider the frictionless, one-dimensional flow of a calorically perfect gas for which there is only a heat interaction with the surroundings and the throughflow area is varied in such a way that Mach number remains constant, i.e., (a) Find an expression for the area ratio Ae/Ai in terms of 3', M, and Tte/Tti = "re. (b) Find an expression for the total pressure ratio Pte/Pti in terms of the same variables.
- Question : 2.16 - Consider the one-dimensional frictional flow of a calorically perfect gas for which there are no energy interactions with the surroundings and the throughflow area is varied in such a way that the velocity remains constant, i.e., Assume that the duct is of circular cross section and that the skin friction shear stress is given by where the skin friction coefficient C I is constant. (a) Show that the duct diameter is given by D = Di + "/M2Cfx (b) Find an expression for the total pressure ratio Pte/Pti in terms of 7, M, Cf, and x/Di.
- Question : 2.17 - Construct the H-K diagram for parameters of your own choice for a: (a) Scramjet (see Fig. 2.19) (b) Ramjet (see Fig. 2.20) Take notes about what lessons you learned along the way.
- Question : 2.18 - Increase your awareness of the H-K diagram by drawing both branches of choked flow for: (a) Isentropic flow (convergent-divergent nozzle) (b) Rayleigh flow (Where does dH/dK = 0?) (c) Fanno flow
- Question : 2.19 - Use HAP(Gas Tables) to show a sequence of oblique shock waves on the H-K diagram for intervals of 1, 2, and 5 deg. for: (a) M1 = 3 (b) M1 = 5 (c) M1 = 10
- Question : 2.2 - Starting from the total pressure MFP [Eq. (2-83)], show that the quantity and draw lines of constant values on the H-K diagram. (a) For flows of constant M, how does ptA vary with r? (b) Prove that the lines are tangent to lines of constant r at M = 1 (c) Describe the physical meaning of these results.
- Question : 2.21 - Starting from the static pressure MFP (Problem 2.12), show that the quantity and draw lines of constant values on the H-K diagram. (a) For flows of constsnt K, how does pA vary with r? (b) For flows of constant M, how does pA vary with r? (c) Describe the physical meaning of these results.
- Question : 2.22 - For the H-K diagram of Fig. 2.26, show that: (a) The minimum rate of change of Mach number with K occurs when K is 0.5. (b) The minimum rate of change of Mach number with K occurs when the Mach number is (c) Explain the physics of this phenomenon.

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